This lesson has a strong emphasis on writing. In what follows I provide some explicit instruction and practice for students to articulate their mathematical understanding through written language.
Prior to this lesson students completed the Q3 Algebra I Common Writing Prompt: Exponential Functions. The common writing prompt is an open response item from a previous MCAS Assessment. The prompt focuses on interpreting and creating exponential functions from a table assessing a relationship between the number of circles (exponential growth) and the diameter of each circle (exponential decay).
I made some minor adaptations to the prompt to align more with Common Core standards around differentiating between linear and exponential functions.
In terms of the mathematics of the lesson, students gain practice with solving systems of equations. At the same time, students gain experience with Math Practice 4 in regards to modeling and applying systems of equations to solve real-world problems.
At this point, students have practiced and completed the Q3 Algebra I Common Writing Prompt: Exponential Functions from a previous class and the reflection on writing portion begins.
I have students reflect on their writing because it is an important and powerful that I find helps students not only identify areas of writing to work on, but also think about specific ways to improve their writing about mathematics.
To open class, students work on completing the Entry Ticket: Q3 Writing Reflection. For this Entry Ticket, students score their own Q3 Algebra I Common Writing Prompt: Exponential Functions (on Exponential Functions) responses based on both the MCAS Scoring Rubric as well as the three components (Content and Evidence, Organization and Mechanics, and Analysis) of the Salem High School Short Response Rubric.
In this section, I review how to analyze arithmetic and geometric sequences using real world examples. The review will follow the Class Notes: Comparing and Contrasting Arithmetic and Geometric Sequences.
As we review the presentation, I ask students to take Two-Column notes and I observe to make sure that they are actively taking notes. My goal is for students to have the opportunity to work with all four domains of language (reading, writing, speaking and listening) as they process and understand the lesson material. During the presentation, I will periodically use Turn and Talks to encourage all students to speak and listen.
During the Guided Practice section, students work in pairs on three main tasks, outlined on the Collaborative Work: Comparing and Contrasting Arithmetic and Geometric Sequences worksheet.
Task 1: Each student independently works on constructing a sequence (one partner creates an arithmetic sequence and the other partner creates a geometric sequence)
Task 2: Partner 1 explains their work to Partner 2 – Partner 2 actively listens and takes notes.
Task 3: Partner 1 and 2 switch roles from Task 2
I do my best to check in and have students show me their sequences prior to moving on to the next section of class. The reason for checking in is I want to be sure students have a sequence that follows the assigned pattern (arithmetic or geometric). Ensuring that students have correct sequences in a way to scaffold the lesson and set students up for richer response for the exit ticket in the next section of the lesson
Students independently work on the Exit Ticket: Comparing and Contrasting Arithmetic and Geometric Sequences. For this exit ticket, students are asked to create a written response to the following prompt:
“Compare and contrast the sequences that you and your partner created. What are the similarities and differences between arithmetic and geometric sequences.”
After class, I assess students’ Exit Ticket based on my high school's short response writing rubric to assess student work.
To conclude class, I create a Venn diagram as a graphic organizer to compare and contrast arithmetic and geometric sequences.
Students participate by adding observations they made when comparing arithmetic and geometric sequences during today's lesson. I facilitate the discussion. At the end, I encourage students to take a picture of Venn diagram to keep as notes for the main ideas learned in class today. Students like to take and share the pictures.
Here are additional practice problems related to interpreting and creating exponential functions: Class Work: Practice Interpreting and Creating Exponential Functions. This resource contains two examples on interpreting and creating exponential functions. The first example is a table on the number of cells after a certain number of cycles of mitosis. I use the example of mitosis because the majority of my students are taking Biology and it is a good review and interdisciplinary application of mathematics. The second example shows the exponential decay of radioactive material over time.
These problems could also be utilized as a homework and/or extra credit assignment.